Modeling and Optimal Control Analysis Applied to Real Cases of COVID-19 Pandemic with Double Dose Vaccination in Ethiopia

Abstract

The novel coronavirus is a recently discovered member of one of the largest families of viruses with symptoms ranging from a simple cold to excruciating respiratory agony. In the present paper, a deterministic mathematical model is formulated to estimate the transmission dynamics of COVID-19 with the inclusion of control strategies like (i) double-dose vaccination, (ii) prevention, and (iii) treatment. In addition, instead of considering all infectious humans as one unit, we separate them into symptomatic and asymptomatic groups, and the impact is analyzed. This separation is meaningful because various reports indicate that the asymptomatic cases will spread the disease more than the symptomatic cases. The model is proved to be mathematically well-posed and biologically meaningful by showing positivity and boundedness of the solution using the appropriate initial conditions. For the reproduction number, a parametric formula is constructed, and also the associated numerical value is calculated from the reported real data in Ethiopia. Moreover, disease-free and endemic equilibria are determined, and their local and global stabilities are discussed using the Lyapunov function technique. These equilibria are found to be locally asymptotically stable if ${\text{R}}_0<1$ and ${\text{R}}_0>1$ , respectively. Following the model fitting and estimation of the parameter values, sensitivity analysis was performed in order to analyze the impact of each parameter on transmission dynamics. In other words, this study can be used to evaluate how major model parameters affect transmission dynamics and control. Utilizing Pontryagin’s maximal principle, the best control measures are implemented with the aim of lowering the burdens associated with infection, prevention, and treatment. To comprehend and visualize the impact of control techniques on the development of the disease and to illustrate the analytical findings generated in this study, numerical simulation studies are conducted. Finally, the output of the study illustrates that adhering to all the control strategies has a big impact on minimizing the transmission of the disease in society. Which means that if the control strategies are well managed by the concerned body, then the burden of the disease is reduced quickly in the Ethiopian population.